Multi-term fractional differential equations offer a more detailed view of processes with different levels of memory and relaxation. Hence, they provide enhanced accuracy in modelling real-world systems in viscoelasticity, finance, control theory, and many other fields. However, the consideration of both the perturbation and the convection terms in this type of differential equations complicates the study of this type of problems. Accordingly, they are usually addressed in the literature using numerical methods, such as the finite difference method. In this work, we investigate a fractional multi-term differential equation where the leading derivative is the Riemann-Liouville- Caputo fractional derivative, combined with separated fractional boundary conditions. In effect, recent applications suggest that this new derivative is be more suitable in certain situations. Moreover, fractional boundary conditions offer a powerful extension of classical ones, but they complicate the solution’s form. Nevertheless, employing fractional order derivatives in boundary conditions is more advantageous, as the latter corresponds to a broader situation, permitting more flexibility in the choice of the derivative, resulting in better outcomes when modelling real-world models in contrast to the classical integer order models. To solve the problem at hand we convert it into a Volterra equation. Then by means of corresponding Green’s functions, we apply fixed point theory techniques to obtain existence results under different conditions imposed on the nonlinearity. Mainly, the Banach contraction principle, the Krasnoselskii fixed point theorem, and the Leray-Schauder nonlinear alternative are utilised to prove the existence of solutions.

This paper investigates the dynamic behavior of a lubricated mechanical system in the incompressible fluid case. The main objective is to develop a rigorous mathematical model describing the motion of the upper surface subjected to lubrication effects. The model is based on the quasistationary Reynolds equation, which governs the pressure distribution within the lubricant film, coupled with Newton’s second law to characterize the vertical displacement of the moving surface.

We establish theoretical results concerning the existence and uniqueness of solutions to the coupled differential system arising from this model. In addition, we identify conditions under which global solutions fail to exist, providing insight into possible mechanical instabilities or breakdown of the lubrication regime.

A detailed numerical analysis is performed to investigate the influence of key physical parameters, particularly the film thickness parameter $a$ and the pressure field $p$, on the overall system dynamics. Various computational experiments are carried out to illustrate the evolution of these parameters and their interaction with the mechanical response of the system.

The obtained results contribute to a deeper understanding of lubricated contact mechanisms and highlight the critical role of fluid-structure interaction in determining system stability. These findings may support the design and optimization of engineering devices involving thin film lubrication and moving rigid surfaces.

Fractional quantum mechanics generalizes standard quantum mechanics to include nonlocality, memory effects, and fractal structures. It retains core quantum characteristics while introducing nonlocal dynamics, which poses substantial challenges for mathematical analysis. The semiclassical analysis, which is among the most significant mathematical frameworks for investigating conventional quantum mechanics, is lacking for fractional quantum mechanics, due to the unmatched difficulties raised by the nonlocal dynamics. In this talk, we will present recent progress toward the development of semiclassical analysis for fractional quantum mechanics, including both analytical and numerical investigations. Semiclassical approximations for the fractional Schrodinger equations will be presented, where ansatz in the form based on Wentzel–Kramers–Brillouin-Jeffreys (WKBJ), Hankel functions, and Fox-H functions are derived. In the semiclassical approximations, the phase and amplitude are proved to be determined by Hamilon-Jacobi type partial differential equations. The semiclassical approximations, as well as the Hamilton-Jacobi type partial differential equations, reduce consistently to those in the semiclassical analysis for standard quantum mechanics when the fractional order approaches integer order, which justifies that the derived semiclassical approximations generalizes those for standard quantum mechanics. The performance of different types of approximations will be compared, showing that the Fox-H function based approximation can achieve uniform accuracy near point sources, in constrast to those based on WKBJ or Hankel functions. Numerical experiments are performed to further justify the derived semiclassical apporixmations.

Matrix-based representations provide a natural and effective way to encode analytic operations arising in generating-function calculus and polynomial systems. In this work, we explore a finite-dimensional matrix-algebraic framework associated with analytic generating kernels of bivariate s–Appell type. By introducing appropriate Wronskian coordinate vectors, we construct families of structured lower-triangular matrices of the Pascal type that act naturally on polynomial coordinate spaces and reflect underlying analytic transformations. This matrix formulation makes it possible to reinterpret classical analytic operations such as differentiation, translation, and recurrence in terms of matrix actions and binomial-type convolutions. Within this unified setting, several structural properties of the associated polynomial sequences, including recurrence relations, shift identities, and differential-type relations, can be systematically examined from a linear-algebraic and operator-theoretic perspective. The approach allows different classes of bivariate polynomial systems, including classical Appell-type constructions and more general systems, to be treated within a common finite-dimensional framework. By organizing generating-function-based structures through matrix algebras, the proposed viewpoint offers a coherent analytical tool for studying polynomial systems and their associated transformations. Overall, this work highlights the usefulness of structured matrix methods in mathematical analysis and demonstrates how finite-dimensional operator representations can provide insight into the analytic behavior and algebraic structure of bivariate polynomial families.

A gene regulatory network is a network consisting of genes that interact with proteins in order to control gene expression. Recent developments in biology and biomedicine have led to the examination of mathematical models that can describe the dynamics of gene regulatory networks.

We present a switched model of gene regulatory networks. The dynamics of both the concentrations of messenger ribonucleic acid and protein are described by ordinary differential equations (ODEs). The set of models with ODE with different activator functions of Hill’s type is given initially. The switching rule is a piecewise constant function and its points of discontinuity determine the points at which the corresponding model from the given set is chosen. An algorithm for the construction of the solution of the switched model and the global equilibrium is presented. Sufficient conditions for the exponential stability of the model are established theoretically. Several examples of the studied switched models are studied. Both cases of a finite and an infinite sequence of switching times are examined. The provided examples validate the obtained sufficient conditions and demonstrate the effect of the presence of the switching rule on the global equilibrium of the model. These examples also show the possibilities of applications of the obtained theoretical results for more adequate modeling of gene regulatory networks.

We investigate a time-dependent abstract Cauchy problem formulated as follows:
\begin{equation}
\begin{aligned}
x'(s) = A(s+t)x(s), \quad t, s \geq 0, \quad x(0) = Cx_0
\end{aligned}
\end{equation}

In this formulation, the function $x(s)$ represents an unknown function defined on the interval $[0,T]$ with values in a Banach space $X$. The operator $C$ is a bounded injective linear operator on $X$, while $A(s)$ denotes a closed linear operator in $X$ with constant domain $\mathcal{D}(A(t))=\mathcal{D}$ for all $t \geq 0$.

The solution to Equation (1) can be formally expressed as $x(t) = U(t,s)x_0$, where $\left\lbrace U(t,s)\right\rbrace_{t,s\geq 0}$ constitutes a two-parameter family of bounded linear operators on $X$, known as a $C$-quasi-semigroup or regularized quasi-semigroup. This concept, introduced by M. Janfada, generalizes classical $C_0$-semigroups to accommodate broader classes of evolution equations.

Our approach extends established spectral techniques from C₀-quasi-semigroups to the general framework of C-quasi-semigroups. The analysis investigates spectral properties of these operator families and their infinitesimal generators, which govern solution behavior.

We establish important spectral inclusion relationships between various spectra of C-quasi-semigroups and their infinitesimal generators. Specifically, we demonstrate spectral inclusions for Saphar, essentially Saphar, quasi-Fredholm, Kato, and essentially Kato spectra, providing comprehensive spectral characterization of these operators.

This work significantly extends the spectral theory of C₀-quasi-semigroups to regularized C-quasi-semigroups. The established spectral inclusion relations offer essential theoretical tools for analyzing regularized operators and their asymptotic properties, opening new perspectives for studying time-dependent differential equations in Banach spaces with regularization techniques.

In this paper, we introduce a new projection-based second-order dynamical system for solving inverse mixed variational inequality problems (IMVIs) with two time-dependent parameters. Such problems arise naturally in optimization and equilibrium theory and are often difficult to handle using standard first-order methods. The proposed dynamical model is designed by combining inertial effects with projection operators, which improves both stability and convergence behavior.

Under the assumptions that the underlying operator is strongly monotone and Lipschitz continuous, we establish the existence and uniqueness of solutions to the proposed system. Moreover, we prove that the equilibrium point of the dynamical system is globally stable. The stability analysis is carried out using an appropriate Lyapunov function, which provides a clear theoretical justification for the convergence of the trajectories.

Additionally, a discrete-time version of the continuous model is derived, resulting in an inertial projection-type iterative algorithm. It is demonstrated that, under suitable parameter choices, the generated sequence converges linearly to the unique solution of the IMVI. This discrete scheme is simple to implement and can be viewed as an efficient numerical realization of the continuous dynamics.

Finally, several numerical experiments are presented to illustrate the effectiveness of the proposed approach. The results confirm the theoretical findings and demonstrate the method's accuracy, fast convergence, and robustness when applied to inverse mixed variational inequalities and related optimization problems.

Solution blow-up is a well-known phenomenon occurring in many Ordinary Differential Equations (ODEs), including ODEs describing real physical systems. For both theoretical and practical (prevention of technogenic catastrophes) reasons, it is important to identify control strategies that dampen blow-up solutions.

First-order non-linear ODEs are used where the Right-Hand Side (RHS) of the equation remains convex and continuous, but otherwise, an arbitrary, non-linear function of unknown solution is considered. RHS also contains a parameter that generally may have different values in different physical circumstances.

A specific control objective (boundedness, at all times, of the solution by a specified threshold) is imposed. This control strategy is designed considering the system parameter as a control variable (practically, variation requires control of the system). The methodology for finding a suitable control consists of constructing a majorizing function. The majorizing function provides an upper bound for the RHS of the equation, and thus, due to the basic comparison theorem of ODEs, the solution of the ODE with the majorizing function on the RHS (the “majorized equation”) provides the bounds for the solution of the original ODE. The majorizing function, utilizing analytical solutions of a majorized equation with specific control variable profiles, is constructed.

A major result is then obtained by applying the comparison theorem of ODEs. This ensures that the solution of the original, un-majorized equation, with an identical control variable profile, also remains bounded over infinitely large time intervals and, moreover, below an a priori-imposed threshold.

Specific results are obtained and considered an example of the control of thermal explosion. Several explicit control variable profiles are considered, and analytical solutions for majorized equations are obtained for all of them. This provides estimations of the bounded behavior of the solutions of the original ODE under identical control conditions.

In conclusion, control strategies, preventing solution blow-up are constructed for a certain class of ODEs.

A geometric algebra (Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. It is built out of two fundamental operations: addition and geometric product. The multiplication of vectors alone results in objects called multivectors, among which are bivectors, the name applied in this paper to the objects of the bivector field ℝ², corresponding to the field of vectors V₂. Compared with other formalisms for manipulating geometric objects, geometric algebra supports dividing by a vector. On the other hand, although rarely used explicitly, a geometric representation of complex numbers is implicitly based on its structure of the Euclidean 2-dimensional vector space. On the basis of the isomorphic algebraic structures of the field of complex numbers ℂ and the 2-dimensional Euclidean field of real vectors V₂, in terms of identical geometric products of elements, integral identities for scalar and vector fields in V₂ are presented, which are vector analogues of the well-known integral identities of complex analysis. Consequtly, these undoubtedly completely new results open up a wide range of applications in all fields where real vector analysis is used, such as quantum physics, classical physics, and so on. Therefore, special attention is paid to the vector analogue of Cauchy's calculus of residues in the field V₂. Finally, at the very end, the algebraic structure of the three-dimensional vector field V₃ is presented, as well as the corresponding fundamental integral identities.

In high-dimensional settings and for time-demanding models, having an efficient approach for computing i) the traditional gradient of every smooth function ($\nabla f$), and ii) the dependent gradient of functions evaluated at non-independent variables ($grad f$) is worth investigating.

In addition to the adjoint methods that provide exact traditional gradients for some classes of PDE/ODE-based models using only one run, this study relies on randomized schemes or the Monte-Carlo approach for computing both gradients. The proposed approach makes use of $\ell_p$-spherical distributions with $p\geq 1$ and Richardson's extrapolation to derive generalized stochastic surrogates of gradients using $L$-point-based evaluations of functions with $L\geq 1$. Such $\ell_p$-spherical-based surrogates of gradients and the corresponding estimators benefit from:

i) Dimension-free upper-bound of the bias;
ii) Dimension-free upper-bounds of mean squared errors (MSEs) and rates of convergence of the form $d^{2/p} N^{-1}$ with $N$ sample size;
iii) Computational efficiency and accuracy.

As a consequence, the proposed approach does not suffer from the drawbacks of dimensionality by properly choosing $p$. It improves the best known rate (i.e., $dN^{-1}$) and enables computations of gradients using a number of function evaluations $N \ll d$ by breaking down the course of dimensionality.

References

[1] O. Shamir, An optimal algorithm for bandit and zero-order convex optimization with two-point feedback, J. Mach. Learn. Res. 18 (1) (2017) 1703#1713.

[2] M. Lamboni, Dimension-free estimators of gradients of functions with(out) non-independent variables, Axioms 15 (1) (2026).

[3] A. Akhavan, E. Chzhen, M. Pontil, A. B. Tsybakov, Gradient-free optimization of highly smooth functions: improved analysis and a new algorithm, Journal of Machine Learning Research 25 (370) (2024) 1#50.

[4] A. S. Berahas, L. Cao, K. Choromanski, K. Scheinberg, A theoretical and empirical comparison of gradient approximations in derivative-free optimization, Foundations of Computational Mathematics 22 (2) (2022) 507#560.